

A308267


Numbers which divide their Zeckendorffian format exactly.


0



1, 2, 4, 7, 8, 14, 16, 28, 31, 32, 56, 62, 64, 83, 112, 124, 127, 128, 166, 224, 248, 254, 256, 332, 397, 448, 496, 508, 511, 512, 664, 794, 891, 896, 992, 1016, 1022, 1024, 1163, 1328, 1588, 1782, 1792, 1984, 2032, 2044, 2047, 2048, 2326, 2656, 3176, 3441, 3564, 3584, 3968
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OFFSET

1,2


COMMENTS

Indices of A048678 which divide into their Zeckendorffian format.
Curiously seems to include the powers of 2, some Mersennes and perfects.
Fixed points of the Zeckendorf function are 2^k.
All numbers 2^(2k+1)1 are in this sequence, and that if n is in this sequence then so is 2n.  Charlie Neder, May 17 2019


LINKS

Table of n, a(n) for n=1..55.


EXAMPLE

The first few terms of A048678 are 1,2,5,4,9,10,21,8. 2 is a multiple of 2, 5 isn't a multiple of 3, 4 is a multiple of 4, 9 isn't a multiple of 5, 10 isn't a multiple of 6, 21 is a multiple of 7, etc.


MATHEMATICA

Select[Range[4000], Divisible[FromDigits[Flatten[IntegerDigits[#, 2] /. {1 > {0, 1}}], 2], #] &] (* Amiram Eldar, Jul 08 2019 after Robert G. Wilson v at A048678 *)


PROG

(Haskell)
import Data.Numbers.Primes
bintodec :: [Int] > Int
bintodec = sum . zipWith (*) (iterate (*2) 1) . reverse
decomp :: (Integer, [Integer]) > (Integer, [Integer])
decomp (x, ys) = if even x then (x `div` 2, 0:ys) else (x  1, 1:ys)
zeck :: Integer > String
zeck n = bintodec (1 : snd (last $ takeWhile (\(x, ys) > x > 0) $ iterate decomp (n, [])))
output :: [Integer]
output = filter (\x > 0 == zeck x `mod` x) [1..100]


CROSSREFS

Cf. A048678, A083420 (subsequence).
Sequence in context: A324588 A201364 A225318 * A177805 A003591 A029746
Adjacent sequences: A308264 A308265 A308266 * A308268 A308269 A308270


KEYWORD

nonn,base


AUTHOR

Dan Dart, May 17 2019


STATUS

approved



